February  2011, 5(1): 137-166. doi: 10.3934/ipi.2011.5.137

A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data

1. 

Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstraβe 69, A-4040 Linz, Austria, Austria

2. 

Institut für Mathematik, Universität Graz, Heinrichstrasse 36, A-8010 Graz, Austria

Received  March 2009 Revised  February 2010 Published  February 2011

This paper presents a level-set based approach for the simultaneous reconstruction and segmentation of the activity as well as the density distribution from tomography data gathered by an integrated SPECT/CT scanner.
   Activity and density distributions are modeled as piecewise constant functions. The segmenting contours and the corresponding function values of both the activity and the density distribution are found as minimizers of a Mumford-Shah like functional over the set of admissible contours and -- for fixed contours -- over the spaces of piecewise constant density and activity distributions which may be discontinuous across their corresponding contours. For the latter step a Newton method is used to solve the nonlinear optimality system. Shape sensitivity calculus is used to find a descent direction for the cost functional with respect to the geometrical variables which leads to an update formula for the contours in the level-set framework. A heuristic approach for the insertion of new components for the activity as well as the density function is used. The method is tested for synthetic data with different noise levels.
Citation: Esther Klann, Ronny Ramlau, Wolfgang Ring. A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. Inverse Problems & Imaging, 2011, 5 (1) : 137-166. doi: 10.3934/ipi.2011.5.137
References:
[1]

G. Aubert and P. Kornprobst, "Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations, with a Foreword by Olivier Faugeras,", Springer-Verlag, (2002).   Google Scholar

[2]

H. B. Ameur, M. Burger and B. Hackl, Level set methods for geometric inverse problems in linear elasticity,, Inverse Problems, 20 (2004), 673.  doi: 10.1088/0266-5611/20/3/003.  Google Scholar

[3]

H. B. Ameur, M. Burger and B. Hackl, Cavity identification in linear elasticity and thermoelasticity,, Math. Methods Appl. Sci., 30 (2007), 625.  doi: 10.1002/mma.772.  Google Scholar

[4]

A. K. Buck, S. Nekolla, S. Ziegler, A. Beer, B. J. Krause, K. Herrmann, K. Scheidhauer, H.-J. Wester, E. J. Rummeny, M. Schwaiger and A. Drzezga, SPECT/CT,, The Journal of Nuclear Medicine, 49 (2008), 1305.   Google Scholar

[5]

J. K. Bucsko, SPECT/CT - The future is clear,, Radiology Today, 5 (2004).   Google Scholar

[6]

M. Burger, Levenberg-Marquardt level set methods for inverse obstacle problems,, Inverse Problems, 20 (2004), 259.  doi: 10.1088/0266-5611/20/1/016.  Google Scholar

[7]

V. Caselles, F. Catté, T. Coll and F. Dibos, A geometric model for active contours in image processing,, Numer. Math., 66 (1993), 1.  doi: 10.1007/BF01385685.  Google Scholar

[8]

A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations,, SIAM J. Appl. Math., 55 (1995), 827.  doi: 10.1137/S0036139993257132.  Google Scholar

[9]

T. F. Chan and L. A. Vese, "Image Segmentation Using Level Sets and the Piecewise Constant Mumford-Shah Model,", UCLA CAM Report 00-14, (2000), 00.   Google Scholar

[10]

T. F. Chan and L. A. Vese, "A Level Set Algorithm for Minimizing the Mumford-Shah Functional in Image Processing,", UCLA CAM Report 00-13, (2000), 00.   Google Scholar

[11]

T. F. Chan and L. A. Vese, Active contours without edges,, IEEE Trans. Image Processing, 10 (2001), 266.  doi: 10.1109/83.902291.  Google Scholar

[12]

T. F. Chan and X.-C. Tai, Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients,, J. Comput. Phys., 193 (2004), 40.  doi: 10.1016/j.jcp.2003.08.003.  Google Scholar

[13]

L. D. Cohen and R. Kimmel, Global minimum for active contour models: A minimum path approach,, International Journal of Computer Vision, 24 (1997), 57.  doi: 10.1023/A:1007922224810.  Google Scholar

[14]

D. Delbeke, R. E. Coleman, M. J. Guiberteau, M. L. Brown, H. D. Royal, B. A. Siegel, D. W. Townsend, L. L. Berland, J. A. Parker, G. Zubal and V. Cronin, Procedure Guideline for SPECT/CT Imaging,, The Journal of Nuclear Medicine, 47 (2006), 1227.   Google Scholar

[15]

M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization,", Society for Industrial and Applied Mathematics (SIAM), (2001).   Google Scholar

[16]

V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography,, Inverse Probl., 15 (1999), 931.  doi: 10.1088/0266-5611/15/4/307.  Google Scholar

[17]

O. Dorn, Shape reconstruction in scattering media with voids using a transport model and level sets,, Can. Appl. Math. Q., 10 (2002), 239.   Google Scholar

[18]

O. Dorn, E. L. Miller and C. M. Rappaport, A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Electromagnetic imaging and inversion of the Earth's subsurface,, Inverse Problems, 16 (2000), 1119.  doi: 10.1088/0266-5611/16/5/303.  Google Scholar

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J.-P. Guillement, F. Jauberteau, L. Kunyansky, R. Novikov and R. Trebossen, On single-photon emission computed tomography imaging based on an exact formula for the nonuniform attenuation correction,, Inverse Probl., 18 (2002).   Google Scholar

[20]

M. Hintermüller and W. Ring, "A Level Set Approach for the Solution of a State Constrained Optimal Control Problem,", Technical Report 212, (2001).   Google Scholar

[21]

M. Hintermüller and W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional,, J. Math. Imag. Vis., 20 (2004), 19.  doi: 10.1023/B:JMIV.0000011317.13643.3a.  Google Scholar

[22]

M. Hintermüller and W. Ring, A second order shape optimization approach for image segmentation,, SIAM J. Appl. Math., 64 (2003), 442.  doi: 10.1137/S0036139902403901.  Google Scholar

[23]

K. Ito, K. Kunisch and Z. Li, Level-set function approach to an inverse interface problem,, Inverse Problems, 17 (2001), 1225.  doi: 10.1088/0266-5611/17/5/301.  Google Scholar

[24]

S. Jehan-Besson, M. Barlaud and G. Aubert, DREAM$^2$S: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation,, November 2001., (2001).   Google Scholar

[25]

S. Jehan-Besson, M. Barlaud and G. Aubert, Video object segmentation using Eulerian region-based active contours,, in, (2001).   Google Scholar

[26]

M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models,, Int. J. of Computer Vision, 1 (1987), 321.  doi: 10.1007/BF00133570.  Google Scholar

[27]

H. Kudo and H. Nakamura, A new appraoch to SPECT attenuation correction without transmission maesurements,, Nuclear Science Symposium Conference Record, 2 (2000), 58.   Google Scholar

[28]

L. A. Kunyansky, A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula,, Inverse Probl., 17 (2001), 293.  doi: 10.1088/0266-5611/17/2/309.  Google Scholar

[29]

A. Litman, D. Lesselier and F. Santosa, Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set,, Inverse Problems, 14 (1998), 685.  doi: 10.1088/0266-5611/14/3/018.  Google Scholar

[30]

M. Mancas, B. Gosselin and B. Macq, Segmentation using a region-growing thresholding,, in, 5672 (2005), 388.   Google Scholar

[31]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math., 42 (1989), 577.  doi: 10.1002/cpa.3160420503.  Google Scholar

[32]

F. Natterer, Inversion of the attenuated Radon transform,, Inverse Probl., 17 (2001), 113.  doi: 10.1088/0266-5611/17/1/309.  Google Scholar

[33]

F. Natterer, "The Mathematics of Computerized Tomography,", Volume \textbf{32} of Classics in Applied Mathematics, 32 (2001).   Google Scholar

[34]

F. Natterer, Determination of tissue attenuation in emission tomography of optically dense media,, Inverse Probl., 9 (1993), 731.  doi: 10.1088/0266-5611/9/6/009.  Google Scholar

[35]

R. G. Novikov, An inversion formula for the attenuated $X$-ray transformation,, Ark. Mat., 40 (2002), 145.  doi: 10.1007/BF02384507.  Google Scholar

[36]

S. J. Osher and R. P. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces,", Springer Verlag, (2002).   Google Scholar

[37]

S. J. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints. I, Frequencies of a two-density inhomogeneous drum,, J. Comput. Phys., 171 (2001), 272.  doi: 10.1006/jcph.2001.6789.  Google Scholar

[38]

N. Paragios and R. Deriche, Geodesic active regions: A new paradigm to deal with frame partition problems in computer vision,, International Journal of Visual Communication and Image Representation, (2001).   Google Scholar

[39]

G. N. Ramachandran and A. V. Lakshminarayanan, Three-dimensional reconstruction from radiographs and electron micrographs: Application of convolutions instead of fourier transforms,, PNAS, 68 (1971), 2236.  doi: 10.1073/pnas.68.9.2236.  Google Scholar

[40]

R. Ramlau, R. Clackdoyle, R. Noo and G. Bal, Accurate attenuation correction in SPECT imaging using optimization of bilinear functions and assuming an unknown spatially-varying attenuation distribution,, ZAMM, 80 (2000), 613.  doi: 10.1002/1521-4001(200009)80:9<613::AID-ZAMM613>3.0.CO;2-9.  Google Scholar

[41]

R. Ramlau, TIGRA-An iterative algorithm for regularizing nonlinear ill-posed problems,, Inverse Probl., 19 (2003), 433.   Google Scholar

[42]

R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data,, J. Comput. Phys., 221 (2007), 539.  doi: 10.1016/j.jcp.2006.06.041.  Google Scholar

[43]

F. Santosa, A level-set approach for inverse problems involving obstacles,, ESAIM: Control, 1 (1996), 17.  doi: 10.1051/cocv:1996101.  Google Scholar

[44]

G. Sapiro, "Geometric Partial Differential Equations and Image Analysis,", Cambridge University Press, (2001).   Google Scholar

[45]

J. A. Sethian, "Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science,", Cambridge University Press, (1999).   Google Scholar

[46]

L. A. Shepp and B. F. Logan, The Fourier reconstruction of a head section,, IEEE Trans. Nucl. Sci, (1974), 21.   Google Scholar

[47]

J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis,", Springer-Verlag, (1992).   Google Scholar

[48]

J. A. Terry, B. M. W. Tsui, J. R. Perry, J. L. Hendricks and G. T. Gullberg, The design of a mathematical phantom of the upper human torso for use in 3-d spect imaging research,, in, (1990), 1467.   Google Scholar

[49]

D. Terzopoulos, Deformable models: Classic, topology-adaptive and generalized formulations,, in, (2003), 21.  doi: 10.1007/0-387-21810-6_2.  Google Scholar

[50]

O. Tretiak and C. Metz, The exponential Radon transform,, SIAM J. Appl. Math., 39 (1980), 341.  doi: 10.1137/0139029.  Google Scholar

[51]

A. Tsai, A. Yezzi and A. S. Willsky, Curve evolution implementation of the Mumford-Shah functional for image segementation, denoising, interpolation, and magnification,, IEEE Transactions on Image Processing, 10 (2001), 1169.  doi: 10.1109/83.935033.  Google Scholar

[52]

J. Weickert, "Anisotropic Diffusion in Image Processing,", European Consortium for Mathematics in Industry, (1998).   Google Scholar

[53]

A. Welch, R. Clack, F. Natterer and G. T. Herman, Toward accurate attenuation correction in SPECT without transmission measurements,, IEEE Trans. Med. Imaging, 16 (1997), 532.  doi: 10.1109/42.640743.  Google Scholar

[54]

M. N. Wernick and J. N. Aarsvold (eds.), "Emission Tomography. The Fundamentals of PET and SPECT,", Elsevier Academic Press, (2004).   Google Scholar

[55]

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen and J. Sikora, Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,, Inverse Probl., 22 (2006), 1509.  doi: 10.1088/0266-5611/22/5/001.  Google Scholar

show all references

References:
[1]

G. Aubert and P. Kornprobst, "Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations, with a Foreword by Olivier Faugeras,", Springer-Verlag, (2002).   Google Scholar

[2]

H. B. Ameur, M. Burger and B. Hackl, Level set methods for geometric inverse problems in linear elasticity,, Inverse Problems, 20 (2004), 673.  doi: 10.1088/0266-5611/20/3/003.  Google Scholar

[3]

H. B. Ameur, M. Burger and B. Hackl, Cavity identification in linear elasticity and thermoelasticity,, Math. Methods Appl. Sci., 30 (2007), 625.  doi: 10.1002/mma.772.  Google Scholar

[4]

A. K. Buck, S. Nekolla, S. Ziegler, A. Beer, B. J. Krause, K. Herrmann, K. Scheidhauer, H.-J. Wester, E. J. Rummeny, M. Schwaiger and A. Drzezga, SPECT/CT,, The Journal of Nuclear Medicine, 49 (2008), 1305.   Google Scholar

[5]

J. K. Bucsko, SPECT/CT - The future is clear,, Radiology Today, 5 (2004).   Google Scholar

[6]

M. Burger, Levenberg-Marquardt level set methods for inverse obstacle problems,, Inverse Problems, 20 (2004), 259.  doi: 10.1088/0266-5611/20/1/016.  Google Scholar

[7]

V. Caselles, F. Catté, T. Coll and F. Dibos, A geometric model for active contours in image processing,, Numer. Math., 66 (1993), 1.  doi: 10.1007/BF01385685.  Google Scholar

[8]

A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations,, SIAM J. Appl. Math., 55 (1995), 827.  doi: 10.1137/S0036139993257132.  Google Scholar

[9]

T. F. Chan and L. A. Vese, "Image Segmentation Using Level Sets and the Piecewise Constant Mumford-Shah Model,", UCLA CAM Report 00-14, (2000), 00.   Google Scholar

[10]

T. F. Chan and L. A. Vese, "A Level Set Algorithm for Minimizing the Mumford-Shah Functional in Image Processing,", UCLA CAM Report 00-13, (2000), 00.   Google Scholar

[11]

T. F. Chan and L. A. Vese, Active contours without edges,, IEEE Trans. Image Processing, 10 (2001), 266.  doi: 10.1109/83.902291.  Google Scholar

[12]

T. F. Chan and X.-C. Tai, Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients,, J. Comput. Phys., 193 (2004), 40.  doi: 10.1016/j.jcp.2003.08.003.  Google Scholar

[13]

L. D. Cohen and R. Kimmel, Global minimum for active contour models: A minimum path approach,, International Journal of Computer Vision, 24 (1997), 57.  doi: 10.1023/A:1007922224810.  Google Scholar

[14]

D. Delbeke, R. E. Coleman, M. J. Guiberteau, M. L. Brown, H. D. Royal, B. A. Siegel, D. W. Townsend, L. L. Berland, J. A. Parker, G. Zubal and V. Cronin, Procedure Guideline for SPECT/CT Imaging,, The Journal of Nuclear Medicine, 47 (2006), 1227.   Google Scholar

[15]

M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization,", Society for Industrial and Applied Mathematics (SIAM), (2001).   Google Scholar

[16]

V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography,, Inverse Probl., 15 (1999), 931.  doi: 10.1088/0266-5611/15/4/307.  Google Scholar

[17]

O. Dorn, Shape reconstruction in scattering media with voids using a transport model and level sets,, Can. Appl. Math. Q., 10 (2002), 239.   Google Scholar

[18]

O. Dorn, E. L. Miller and C. M. Rappaport, A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Electromagnetic imaging and inversion of the Earth's subsurface,, Inverse Problems, 16 (2000), 1119.  doi: 10.1088/0266-5611/16/5/303.  Google Scholar

[19]

J.-P. Guillement, F. Jauberteau, L. Kunyansky, R. Novikov and R. Trebossen, On single-photon emission computed tomography imaging based on an exact formula for the nonuniform attenuation correction,, Inverse Probl., 18 (2002).   Google Scholar

[20]

M. Hintermüller and W. Ring, "A Level Set Approach for the Solution of a State Constrained Optimal Control Problem,", Technical Report 212, (2001).   Google Scholar

[21]

M. Hintermüller and W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional,, J. Math. Imag. Vis., 20 (2004), 19.  doi: 10.1023/B:JMIV.0000011317.13643.3a.  Google Scholar

[22]

M. Hintermüller and W. Ring, A second order shape optimization approach for image segmentation,, SIAM J. Appl. Math., 64 (2003), 442.  doi: 10.1137/S0036139902403901.  Google Scholar

[23]

K. Ito, K. Kunisch and Z. Li, Level-set function approach to an inverse interface problem,, Inverse Problems, 17 (2001), 1225.  doi: 10.1088/0266-5611/17/5/301.  Google Scholar

[24]

S. Jehan-Besson, M. Barlaud and G. Aubert, DREAM$^2$S: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation,, November 2001., (2001).   Google Scholar

[25]

S. Jehan-Besson, M. Barlaud and G. Aubert, Video object segmentation using Eulerian region-based active contours,, in, (2001).   Google Scholar

[26]

M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models,, Int. J. of Computer Vision, 1 (1987), 321.  doi: 10.1007/BF00133570.  Google Scholar

[27]

H. Kudo and H. Nakamura, A new appraoch to SPECT attenuation correction without transmission maesurements,, Nuclear Science Symposium Conference Record, 2 (2000), 58.   Google Scholar

[28]

L. A. Kunyansky, A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula,, Inverse Probl., 17 (2001), 293.  doi: 10.1088/0266-5611/17/2/309.  Google Scholar

[29]

A. Litman, D. Lesselier and F. Santosa, Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set,, Inverse Problems, 14 (1998), 685.  doi: 10.1088/0266-5611/14/3/018.  Google Scholar

[30]

M. Mancas, B. Gosselin and B. Macq, Segmentation using a region-growing thresholding,, in, 5672 (2005), 388.   Google Scholar

[31]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math., 42 (1989), 577.  doi: 10.1002/cpa.3160420503.  Google Scholar

[32]

F. Natterer, Inversion of the attenuated Radon transform,, Inverse Probl., 17 (2001), 113.  doi: 10.1088/0266-5611/17/1/309.  Google Scholar

[33]

F. Natterer, "The Mathematics of Computerized Tomography,", Volume \textbf{32} of Classics in Applied Mathematics, 32 (2001).   Google Scholar

[34]

F. Natterer, Determination of tissue attenuation in emission tomography of optically dense media,, Inverse Probl., 9 (1993), 731.  doi: 10.1088/0266-5611/9/6/009.  Google Scholar

[35]

R. G. Novikov, An inversion formula for the attenuated $X$-ray transformation,, Ark. Mat., 40 (2002), 145.  doi: 10.1007/BF02384507.  Google Scholar

[36]

S. J. Osher and R. P. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces,", Springer Verlag, (2002).   Google Scholar

[37]

S. J. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints. I, Frequencies of a two-density inhomogeneous drum,, J. Comput. Phys., 171 (2001), 272.  doi: 10.1006/jcph.2001.6789.  Google Scholar

[38]

N. Paragios and R. Deriche, Geodesic active regions: A new paradigm to deal with frame partition problems in computer vision,, International Journal of Visual Communication and Image Representation, (2001).   Google Scholar

[39]

G. N. Ramachandran and A. V. Lakshminarayanan, Three-dimensional reconstruction from radiographs and electron micrographs: Application of convolutions instead of fourier transforms,, PNAS, 68 (1971), 2236.  doi: 10.1073/pnas.68.9.2236.  Google Scholar

[40]

R. Ramlau, R. Clackdoyle, R. Noo and G. Bal, Accurate attenuation correction in SPECT imaging using optimization of bilinear functions and assuming an unknown spatially-varying attenuation distribution,, ZAMM, 80 (2000), 613.  doi: 10.1002/1521-4001(200009)80:9<613::AID-ZAMM613>3.0.CO;2-9.  Google Scholar

[41]

R. Ramlau, TIGRA-An iterative algorithm for regularizing nonlinear ill-posed problems,, Inverse Probl., 19 (2003), 433.   Google Scholar

[42]

R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data,, J. Comput. Phys., 221 (2007), 539.  doi: 10.1016/j.jcp.2006.06.041.  Google Scholar

[43]

F. Santosa, A level-set approach for inverse problems involving obstacles,, ESAIM: Control, 1 (1996), 17.  doi: 10.1051/cocv:1996101.  Google Scholar

[44]

G. Sapiro, "Geometric Partial Differential Equations and Image Analysis,", Cambridge University Press, (2001).   Google Scholar

[45]

J. A. Sethian, "Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science,", Cambridge University Press, (1999).   Google Scholar

[46]

L. A. Shepp and B. F. Logan, The Fourier reconstruction of a head section,, IEEE Trans. Nucl. Sci, (1974), 21.   Google Scholar

[47]

J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis,", Springer-Verlag, (1992).   Google Scholar

[48]

J. A. Terry, B. M. W. Tsui, J. R. Perry, J. L. Hendricks and G. T. Gullberg, The design of a mathematical phantom of the upper human torso for use in 3-d spect imaging research,, in, (1990), 1467.   Google Scholar

[49]

D. Terzopoulos, Deformable models: Classic, topology-adaptive and generalized formulations,, in, (2003), 21.  doi: 10.1007/0-387-21810-6_2.  Google Scholar

[50]

O. Tretiak and C. Metz, The exponential Radon transform,, SIAM J. Appl. Math., 39 (1980), 341.  doi: 10.1137/0139029.  Google Scholar

[51]

A. Tsai, A. Yezzi and A. S. Willsky, Curve evolution implementation of the Mumford-Shah functional for image segementation, denoising, interpolation, and magnification,, IEEE Transactions on Image Processing, 10 (2001), 1169.  doi: 10.1109/83.935033.  Google Scholar

[52]

J. Weickert, "Anisotropic Diffusion in Image Processing,", European Consortium for Mathematics in Industry, (1998).   Google Scholar

[53]

A. Welch, R. Clack, F. Natterer and G. T. Herman, Toward accurate attenuation correction in SPECT without transmission measurements,, IEEE Trans. Med. Imaging, 16 (1997), 532.  doi: 10.1109/42.640743.  Google Scholar

[54]

M. N. Wernick and J. N. Aarsvold (eds.), "Emission Tomography. The Fundamentals of PET and SPECT,", Elsevier Academic Press, (2004).   Google Scholar

[55]

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen and J. Sikora, Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,, Inverse Probl., 22 (2006), 1509.  doi: 10.1088/0266-5611/22/5/001.  Google Scholar

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